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# Chaos on the Circle MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'],['$$','$$']]}});

by

Professor Emeritus Susan Niefield
Union College

February 8, 2016
5:00PM
Bailey Hall 207

Refreshments will be served in Bailey Hall 204 4:45pm

## Abstract:

Take a point $P$ on a circle in the plane and double the angle $\theta$ to get a point $f(P)$.

Repeating this process gives a set $\{P,f(P),f(f(P)),\dots \}$ of points, called the orbit of $P$ under $f$}. What kind of orbits can we find? Of course, that depends on the starting point $P$. Some orbits are finite, while others are not. Among the infinite ones, there are even orbits that hit almost every point on the circle.

This map is an example of a chaotic dynamical system. After presenting a definition of chaos, we will show that points on a circle can be represented as binary sequences, and use this representation to prove that the angle-doubling map $f$ is chaotic.

For additional information, send e-mail to math@union.edu or call (518) 388-6246.
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